Let $\mathcal{l}$  be a line which is normal to the curve y = 2x² + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line $\mathcal{l}$  and O is origin, then the area of the triangle OPQ is equal to

$y=2x^2+x+2....(i)$

$\frac{dy}{dx}=4x+1$

Slope of normal

$-\frac{dx}{dy}=-\frac{1}{4x+1}$

Equation of PQ y - β = $-\frac{1}{4\alpha +1}\left(x-\alpha \right)$

It passes (6, 4)

$\Rightarrow \left(4-\beta \right)\left(4\alpha +1\right)=-\left(6-\alpha \right)$

$4{\alpha }^3+3{\alpha }^2-3\alpha -3=0$

<div><picture><img src="https://images.shiksha.com/mediadata/images/1752484260php26Cqhb.jpeg" alt="" width="273" height="209" loading="lazy"></picture></div><div><p><math><mrow><mi>y</mi><mo>=</mo><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false"> (</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math></p><p><math><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></p><p>Slope of normal</p><p><math><mrow><mo>−</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></p><p>Equation of PQ y - β = <math><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mi>α</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mrow><mo> (</mo><mrow><mi>x</mi><mo>−</mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></math></p><p>It passes (6, 4)</p><p><math><mrow><mo>⇒</mo><mrow><mo> (</mo><mrow><mn>4</mn><mo>−</mo><mi>β</mi></mrow><mo>)</mo></mrow><mrow><mo> (</mo><mrow><mn>4</mn><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mrow><mo> (</mo><mrow><mn>6</mn><mo>−</mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></math></p><p><math><mrow><mn>4</mn><msup><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>3</mn><msup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>3</mn><mi>α</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></p></div>

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